Question: Dominique from "Dominique's Pizza" bakes $p$ pizzas every day. Currently, it costs her $\$8$ per day to use the oven and $\$1.50$ per pizza for the ingredients. Tomorrow, the price for the ingredients will increase from $\$1.50$ per pizza to $\$2$ per pizza. The oven costs will stay the same at $\$8$ per day. Dominique did some calculations and found that she should bake $8$ more pizzas each day in order for the total expenses per pizza (including ingredient and shared oven costs) to remain the same. Write an equation in terms of $p$ to model the situation.
Solution: The strategy Dominique did some calculations to ensure that her total per pizza expenses (including ingredients and shared oven costs) will be the same before and after the ingredient prices increase. If we let $C$ be the current per pizza expense and $T$ be tomorrow's per pizza expense, then we obtain the equation $C=T$. Now, let's express $C$ and $T$ in terms of $p$. Expressing the current per pizza costs Currently, Dominique pays a flat fee of $\$8$ every day to use the oven. Dominique bakes $p$ pizzas every day, and the ingredients cost $\$1.50$ per pizza. Therefore, $1.50p$ dollars represents the total cost that Dominique must pay for ingredients currently. Adding this to the $\$8$ flat fee, we see that $8+1.50p$ dollars represents the total per day cost to bake $p$ pizzas. To find the per pizza cost, we can divide the total cost by the number of pizzas she makes. So the current total expense per pizza is $\:\dfrac{8+1.50p}{p}$ dollars. Expressing tomorrow's per pizza costs Tomorrow, Dominique will still pay a flat fee of $\$8$ to use the oven. However, Dominique will now bake $p+8$ pizzas and the ingredients will cost $\$2$ per pizza. Therefore, $2(p+8)$ dollars represents the total cost that Dominique must pay for pizza ingredients tomorrow. Adding this to the $\$8$ flat fee, we see that $8+2(p+8)$ dollars represents the total per day cost to bake $p+8$ pizzas. To find the per pizza cost, we can divide the total cost by the number of pizzas she makes. So tomorrow's total expense per pizza is $\:\dfrac{8+2(p+8)}{p+8}$ dollars or $\dfrac{2p+24}{p+8}$ dollars. Putting things together We found that $C=\dfrac{8+1.50p}{p}$ and $T=\dfrac{2p+24}{p+8} $. Since $C=T$, we can substitute and find an equation in terms of $p$ that models the situation. The answer is: $ \dfrac{8+1.50p}{p}=\dfrac{2p+24}{p+8} $